CDS Mathematics · Circles

Tangents from an External Point

A tangent is perpendicular to the radius at its point of contact, and the two tangents from one outside point are equal.

All Circles notes CDS Mathematics strategy

Why this matters

Eleven PYQs, none HARD. Three facts carry the page: tangent ⊥ radius (a right triangle with the line to the centre), equal tangents from one point, and the alternate segment theorem for the angle a tangent makes with a chord.

Concept 1 of 3: Length of a tangent

The radius to the point of contact, the tangent, and the line from the outside point to the centre form a right triangle, with the right angle at the point of contact.

Definition

Tangent $\perp$ radius at the point of contact: $PT = \sqrt{OP^2 - r^2}$ .
The two tangents from $P$ are equal, and $OP$ bisects the angle between them.
The angle between the tangents and the angle between the radii to the contact points add to $180^\circ$ .
$OP$ is the perpendicular bisector of the chord of contact $MN$ , meeting it at $Q$ with $OM^2 = OQ \cdot OP$ .

Tangent length

PT = \sqrt{OP^2 - r^2}

Worked example

A point is

13

cm from the centre of a circle of radius

5

cm. Find the length of the tangent from it.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q59Moderate

Example 1 · Circles · Tangents from an External Point

If two tangents inclined at an angle

60^\circ

are drawn to a circle of radius 3 cm, then what is the length of each tangent ?

Supplement, not double

The angle between the radii to the contact points is

180^\circ

minus the angle between the tangents, because the quadrilateral has two right angles. It is not twice that angle.

Drill 5 more on length of a tangent

Concept 2 of 3: The alternate segment theorem

Turn a chord

PQ

until

Q

slides onto

P

: the chord becomes the tangent. The angle between a tangent and a chord is the limit of an inscribed angle, so it equals the inscribed angle on the other side.

Definition

The angle between a tangent and a chord through the point of contact equals the inscribed angle in the ALTERNATE segment.
So the central angle on that chord is twice the tangent–chord angle.
Two tangents $XA, XB$ with $\angle AXB = \phi$ : $\angle XAB = \angle XBA = 90^\circ - \dfrac\phi2$ , which is also the angle $AB$ subtends on the far arc.

Tangent–chord angle

\angle QPT = \angle PRQ = \tfrac12\angle POQ

Worked example

The tangent at

P

makes

40^\circ

with the chord

PQ

. Find the angle

PQ

subtends at the centre.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q63Moderate

Example 2 · Circles · Tangents from an External Point

A triangle PQR is inscribed in a circle with its centre at O. A tangent PT is drawn at P such that

\angle QPT = 36^\circ

. What is

\angle POQ

equal to ?

The ALTERNATE segment

The equal inscribed angle is on the far side of the chord from the tangent–chord angle. An angle in the near segment is its supplement.

Drill 2 more on the alternate segment theorem

Concept 3 of 3: A circle inside a quadrilateral

Each vertex sends two equal tangents to the circle. Each pair of opposite sides uses one tangent from every vertex, so the two pairs have the same total.

Definition

If a circle touches all four sides of $ABCD$ : $AB + CD = BC + DA$ .
The perimeter is then $2(AB + CD)$ , so either opposite-pair sum fixes it.
The same equal-tangent idea splits a triangle's sides at the incircle: the tangent from $A$ is $s - a$ .

Tangential quadrilateral

AB + CD = BC + DA

Worked example

A circle touches all four sides of

ABCD

, with

AB = 6

BC = 7

and

CD = 9

cm. Find

DA

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q90Easy

Example 3 · Circles · Tangents from an External Point

A circle touches all the four sides of a quadrilateral

ABCD

. If

AB = 9

cm,

BC = 8

cm and

CD = 12

cm, then what is

DA

equal to?

Opposite sides, not adjacent ones

The rule pairs

AB

with

CD

and

BC

with

DA

. Adding adjacent sides gives nothing.

Drill 1 more on a circle inside a quadrilateral

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Length of a tangent
Tangent length
$PT = \sqrt{OP^2 - r^2}$
The alternate segment theorem
Tangent–chord angle
$\angle QPT = \angle PRQ = \tfrac12\angle POQ$
A circle inside a quadrilateral
Tangential quadrilateral
$AB + CD = BC + DA$

Watch out for (3)

Supplement, not double
→ Length of a tangent
The ALTERNATE segment
→ The alternate segment theorem
Opposite sides, not adjacent ones
→ A circle inside a quadrilateral

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 11 questions